Diffraction of sound through doorway

Diffraction of Sound

Sound with a frequency 650 HZ from a distant source passes through a doorway 1.10 m wide in a sound absorbing wall. Find the number and angular directions of the diffraction minimum at listening positions along a line parallel to the wall.

diffraction of sound
Answer

The doorway can be regarded as a single slit diffraction. For the single slit it is considered that the interference of waves takes place between waves emitted by a point in the middle of the slit opening and one point at the end of the slit opening.

The condition for maximum of interference of waves at the listening positions is
$d*sin(\theta) = k*\lambda$
where $d = 1.1 / 2 =0.55 m$ is the distance between the two wave emitting sources  and $\theta$ is the angle of diffraction

$\lambda = V*T = V/F = 343 / 650 =0.528 m$  is the wavelength
$V =343 m/s$ is speed of sound in air at 20 deg Celsius
Thus  $sin(\theta) =k*\lambda/d$

Number and angular directions of the diffraction minimum

for $\theta = 90 deg$ we have
$k =d/\lambda = int (0.55/0.528) =int(1.04) =1$
(where $int$ is the integer part)
there are two directions for maximum, one for $k=0$ and one for $k=1$
there is one direction for minimum for $k=0$ between the two maximum

We find the the angular direction of minimum from the minimum direction of the diffraction of sound:
$sin(\theta) = (k+1/2)*\lambda/d = 0.5*0.528/0.55 =0.48$
$\theta = 28.68 degree$

Observation:

This is valid if you start counting from middle position of the wall (the position that corresponds to the door opening) towards one end of the wall. If the wall is symmetric with respect to the door, you will have twice as many minima. One at $+28.68 degree$ and one at $-28.68 degree$ from the middle line of the wall.

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